I asked a version of this question on Math Stack Exchange 6 days ago, but without any responses: The area of a certain region
I am interested in evaluating the area of the region defined by
$$A_{L_1, L_2, F} = \left \{(x,y) \in \mathbb{R}^2 : \left \lvert L_1(x,y) L_2(x,y) F(x,y)^2 \right \rvert \leq 1 \right\}$$
where $L_1, L_2$ are linearly independent forms and $F$ is a positive definite quadratic form.
If the square was not there, then the answer of this question can be given explicitly in terms of elliptic integrals; see my paper On binary cubic and quartic forms and this paper of Bean: The practical computation of areas associated with binary quartic forms.
Specifically, I am interested in the case when $L_1(x,y) = y, L_2(x,y) = x + 7y, F(x,y) = x^2 + xy + 7y^2$.
